THE VOLUME CONJECTURE OF KNOTS AND ITS RAMIFICATIONS

结的体积猜想是由Kashaev， J. Murakami和H. Murakami提出的结补的简单体积是由著名的琼斯多项式的推广——有色琼斯多项式的渐近性质决定的。现在，这个猜想通过H. Murakami、J. Murakami、M. Okamoto、T. Takata等人的计算机实验推广到涉及chen - simons不变量，许多数学家对这个问题很感兴趣。本研究的目的是探讨结补几何与有色琼斯多项式之间的关系，我们已经证明了有色琼斯多项式在Neuman-Zagler-Yoshida函数和a -多项式中都占主导地位。实际上，通过对体积猜想的研究，我们从有色琼斯多项式中导出了一个势函数，该势函数可以变形为Neuman-Zagler-Yoshida函数，其偏微分方程给出了a -多项式。在尖头双曲流形的双曲结构的变形空间上定义了Neuman-Zagler-Yoshida函数，描述了体积与chen - simons不变量之间的解析关系。结点的a -多项式是从结点基本群的表示空间中定义的，在Dehn手术理论中起着重要的作用。最近，一些数论学者也对a多项式产生了兴趣，因为它的马勒测度给出了一定的Dedekind zeta函数。作者被邀请参加了许多会议，并有很多机会展示我们的成果。作者希望许多数学家对这一研究感兴趣。

The volume conjecture of knots states that the asymptotic behavior of the colored Jones polynomials, a generalization of the famous Jones polynomial, determines the simplicial volume of the complement of a knot, which is proposed by Kashaev, J. Murakami and H. Murakami. This conjecture is now generalized to involve the Chern-Simons invariant through the computer experiment by H. Murakami, J. Murakami, M. Okamoto, T. Takata and the author, and many mathematicians are now interested in this problem.The purpose of this research is to investigate the relationship between the geometry of the knot complement and the colored Jones polynomials, and we have shown that the colored Jones polynomials dominate both the Neuman-Zagler-Yoshida function and the A-polynomial. In fact, from the study of the volume conjecture, we derive a potential function from the colored Jones polynomials which can be deformed into the Neuman-Zagler-Yoshida function and whose partial differential equations gives the A-polynomial.The Neuman-Zagler-Yoshida function is defined over deformation space of the hyperbolic structures of cusped hyperbolic manifolds and describes the analytic relation between the volumes and the Chern-Simons invariants. The A-polynomial of knots is defined from the representation space of the fundamental groups of knots and plays an important role in the theory of Dehn surgery. Recently, some number theorists are also interested in the A-polynomial because its Mahler measure gives certain Dedekind zeta function.The author was invited to many conferences and had many opportunities to exhibit our results. The author hopes many mathematicians are interested in this research.